True Equation: Identify & Solve | Math Examples

by Admin 48 views
True Equation: Identify & Solve

Hey guys! Let's dive into a fun math problem where we need to figure out which of the following mathematical sentences represents a true equation. We've got some options here: a) 3x + 5 = 20, b) 2y - 7 > 10, c) 4z + 2 ≤ 8, and d) 5a + 3 = 3a + 7. Buckle up, because we're not just picking an answer; we're going to break down what makes an equation an equation and why our chosen answer fits the bill. So, let's get started and make math a little less mysterious!

Understanding Equations

Before we jump into the options, let's make sure we're all on the same page about what an equation actually is. At its heart, an equation is a mathematical statement that asserts the equality of two expressions. Think of it like a balanced scale: what's on one side must be equal to what's on the other. This equality is denoted by the equals sign (=). So, if you see that sign, you know you're dealing with an equation.

Now, the expressions on either side of the equals sign can be simple numbers, or they can be more complex, involving variables, constants, and mathematical operations. The key thing is that the equation is making a claim that, for some value(s) of the variable(s), the two expressions are equivalent. This is where the idea of "solving" an equation comes in – we're trying to find the value(s) of the variable(s) that make the equation true. Equations are fundamental in algebra and are used to model and solve a wide variety of problems in mathematics, science, engineering, and beyond. The equals sign is the glue that holds the equation together, stating that the quantities on either side are the same. Equations can be simple or complex, linear or nonlinear, but the defining characteristic is always the presence of the equals sign.

Key Characteristics of an Equation

  1. Equality Sign: The most important characteristic of an equation is the presence of an equals sign (=). This sign indicates that the expressions on both sides are equal.
  2. Expressions: Equations contain mathematical expressions on both sides of the equals sign. These expressions can include numbers, variables, and mathematical operations.
  3. Variables (Optional): Many equations contain variables, which are symbols (usually letters) that represent unknown values. Solving the equation means finding the value(s) of the variable(s) that make the equation true.
  4. Constants: Constants are fixed numerical values in the equation.
  5. Mathematical Operations: Equations can involve various mathematical operations such as addition, subtraction, multiplication, division, exponentiation, etc.
  6. Solution: The solution to an equation is the value or set of values that, when substituted for the variable(s), makes the equation a true statement.

Analyzing the Options

Alright, now that we know what an equation is, let's examine the options we've been given and see which one fits the definition and has the potential to be a true equation.

  • a) 3x + 5 = 20: This looks promising! We have an equals sign, expressions on both sides, a variable (x), and constants. It certainly appears to be an equation.
  • b) 2y - 7 > 10: Ah, hold on a sec. This one has a "greater than" sign (>). That means it's an inequality, not an equation. Inequalities express a relationship where one side is not necessarily equal to the other. So, b is out.
  • c) 4z + 2 ≤ 8: Similar to option b, this one uses a "less than or equal to" sign (≤). This also makes it an inequality, not an equation. Eliminate c.
  • d) 5a + 3 = 3a + 7: This one's interesting. We have an equals sign, expressions on both sides, a variable (a), and constants. It definitely looks like an equation, and it has the potential to be true depending on the value of 'a'.

The Correct Answer and Why

So, we've narrowed it down to options a) and d). Both look like equations, but the question asks for a true equation. What does that mean? It means that the equation must have a solution—a value for the variable that makes the equation a true statement.

Let's take a closer look at both options:

  • a) 3x + 5 = 20: We can solve this equation for x: 3x = 20 - 5 3x = 15 x = 5 So, when x = 5, the equation is true. This confirms that it's a valid equation.
  • d) 5a + 3 = 3a + 7: We can also solve this equation for a: 5a - 3a = 7 - 3 2a = 4 a = 2 So, when a = 2, the equation is true. This confirms that it's also a valid equation.

However, the question implies there is only one correct answer. The best answer is d) 5a + 3 = 3a + 7. This is because option a) could be seen as an expression, whereas option d) has variables on both sides of the equation, making it a more robust equation in the traditional sense. Option d showcases a balanced equation where simplifying both sides leads to a unique solution for 'a'.

Why Other Options Are Incorrect

To really nail this down, let's reiterate why the other options don't work:

  • b) 2y - 7 > 10: This is an inequality, not an equation. It states that 2y - 7 is greater than 10, not equal to it.
  • c) 4z + 2 ≤ 8: This is also an inequality. It states that 4z + 2 is less than or equal to 8, not equal to it.

Equations always use an equals sign to show that two expressions are equivalent. Inequalities, on the other hand, use signs like >, <, ≥, or ≤ to show that two expressions are not equal but have a specific relationship (greater than, less than, etc.). Inequalities are used to express relationships where one quantity is greater than, less than, greater than or equal to, or less than or equal to another quantity.

Conclusion

So, after carefully examining all the options and understanding what makes an equation an equation, we can confidently say that d) 5a + 3 = 3a + 7 represents a true equation. We were able to solve for 'a' and find a value that makes the equation true. Remember, the key to identifying equations is looking for the equals sign and expressions on both sides that can be equal for some value(s) of the variable(s). Keep practicing, and you'll become a master at spotting and solving equations in no time! Understanding equations is crucial for anyone studying algebra or any field that requires mathematical modeling. The ability to recognize and solve equations opens up a wide range of problem-solving opportunities.